{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-lagrange-theorem-finite-groups",
  "semantic_family_id": "sf-lagrange-theorem-finite-groups",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群的Lagrange定理"
  },
  "summary": {
    "zh-Hant": "有限群G的子群H以等大左陪集分割G，因此|G|=[G:H]|H|且|H|整除|G|。"
  },
  "statement": {
    "zh-Hant": "H須由完整Cayley table認證為subgroup。全部left cosets gH必須相等或不交並完整覆蓋G；h↦gh給出H到gH的bijection，所以每個coset大小為|H|。結論|G|=[G:H]|H|。本輪converse family只證d整除|G|不表示任意指定d-element subset是subgroup；不以該witness證明或否定某個order-d subgroup是否存在。"
  },
  "formula": {
    "tex": "c(G)=i(G,H)*c(H)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mrow><mi>i</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow><mo>×</mo><mrow><mi>c</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "c",
          "glyph": "c"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "G",
            "glyph": "G"
          }
        ]
      },
      "rhs": {
        "type": "multiplication",
        "operands": [
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "i",
              "glyph": "i"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "G",
                "glyph": "G"
              },
              {
                "type": "symbol",
                "name": "H",
                "glyph": "H"
              }
            ]
          },
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "c",
              "glyph": "c"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "H",
                "glyph": "H"
              }
            ]
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "25bf22b5114db935b2320f2dfbc55d2bf027f49322c90b3267fa529e5c98b2a4"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限Cayley table、identity與candidate subset。"
    },
    {
      "id": "a2",
      "zh-Hant": "正向families完整驗identity/product/inverse closure及全部cosets。"
    }
  ],
  "symbols": [
    {
      "token": "c",
      "role": "finite_cardinality",
      "role_zh": "有限基數",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_group",
      "role_zh": "有限群",
      "scope": "representative_formula"
    },
    {
      "token": "i",
      "role": "subgroup_index",
      "role_zh": "子群index",
      "scope": "representative_formula"
    },
    {
      "token": "H",
      "role": "certified_subgroup",
      "role_zh": "已認證子群",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group",
      "title_zh": "有限群",
      "reason_zh": "完整Cayley table。"
    },
    {
      "id": "mko-subgroup-homomorphism",
      "title_zh": "子群與群同態",
      "reason_zh": "subgroup closure。"
    },
    {
      "id": "mko-natural-divisibility",
      "title_zh": "自然數整除關係",
      "reason_zh": "order divisibility。"
    },
    {
      "id": "mko-cardinality-comparison",
      "title_zh": "基數比較",
      "reason_zh": "finite equal-size partition。"
    },
    {
      "id": "mko-quotient-set",
      "title_zh": "商集合",
      "reason_zh": "left-coset quotient partition。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "left translation是H到gH的bijection。",
      "cosets equal or disjoint並覆蓋G。",
      "整除結論沒有一般converse。"
    ],
    "common_misconception_zh": "只因指定subset大小整除群階便稱其為subgroup，或把subset反例誤報成同階subgroup不存在的證書。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪限完整有限群表；不由sampled products證subgroup。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入有限群Cauchy定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-lagrange-theorem-finite-groups.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-11 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}